More Related Content More from Nigel Simmons (20) Binomial Theorem Expansions6. Binomial Theorem
Binomial Expansions
A binomial expression is one which contains two terms.
1 x 0 1
1 x 1 1 1x
1 x 2 1 2 x 1x 2
1 x 1 x 1 2 x 1x 2
3
1 2 x x 2 x 2 x 2 x3
1 3x 3x 2 x 3
7. Binomial Theorem
Binomial Expansions
A binomial expression is one which contains two terms.
1 x 0 1
1 x 1 1 1x
1 x 2 1 2 x 1x 2
1 x 1 x 1 2 x 1x 2
3
1 2 x x 2 x 2 x 2 x3
1 3x 3x 2 x 3
1 x 4 1 x 1 3x 3x 2 x 3
1 3x 3x 2 x 3 x 3x 2 3x 3 x 4
1 4 x 6 x 2 4 x3 x 4
12. Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
1
1 1
1 2 1
1 3 3 1
13. Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
14. Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
15. Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
16. Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
17. Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
18. Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
19. Blaise Pascal saw a pattern which we now know as Pascal’s Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 28 856 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1
20. 7
2x
e.g .i 1
3
21. 7
2x
e.g .i 1
3
2 3 4 5
17 71 211 351 351 211
6 2x 5 2x 4 2x 3 2x 2 2x
3 3 3 3 3
6 7
2x 2x
71
3 3
22. 7
2x
e.g .i 1
3
2 3 4 5
17 71 211 351 351 211
6 2x 5 2x 4 2x 3 2x 2 2x
3 3 3 3 3
6 7
2x 2x
71
3 3
14 x 84 x 2 280 x 3 560 x 4 672 x 5 448 x 6 128 x 7
1
3 9 27 81 243 729 2187
23. 7
2x
e.g .i 1
3
2 3 4 5
17 71 211 351 351 211
6 2x 5 2x 4 2x 3 2x 2 2x
3 3 3 3 3
6 7
2x 2x
71
3 3
14 x 84 x 2 280 x 3 560 x 4 672 x 5 448 x 6 128 x 7
1
3 9 27 81 243 729 2187
ii Use the expansion of 1 x 10 to find the value of 0.99810 to 8 dps
24. 7
1 2 x
e.g .i
3
2 3 4 5
17 71 211 351 351 211
6 2x 5 2x 4 2x 3 2x 2 2x
3 3 3 3 3
6 7
2x 2x
71
3 3
14 x 84 x 2 280 x 3 560 x 4 672 x 5 448 x 6 128 x 7
1
3 9 27 81 243 729 2187
ii Use the expansion of 1 x 10 to find the value of 0.99810 to 8 dps
1 x 10 1 10 x 45 x 2 120 x 3 210 x 4 252 x 5 210 x 6 120 x 7 45 x8
10 x 9 x10
25. 7
1 2 x
e.g .i
3
2 3 4 5
17 71 211 351 351 211
6 2x 5 2x 4 2x 3 2x 2 2x
3 3 3 3 3
6 7
2x 2x
71
3 3
14 x 84 x 2 280 x 3 560 x 4 672 x 5 448 x 6 128 x 7
1
3 9 27 81 243 729 2187
ii Use the expansion of 1 x 10 to find the value of 0.99810 to 8 dps
1 x 10 1 10 x 45 x 2 120 x 3 210 x 4 252 x 5 210 x 6 120 x 7 45 x8
10 x 9 x10
0.99810 1 100.002 450.0022 1200.0023
26. 7
1 2 x
e.g .i
3
2 3 4 5
17 71 211 351 351 211
6 2x 5 2x 4 2x 3 2x 2 2x
3 3 3 3 3
6 7
2x 2x
71
3 3
14 x 84 x 2 280 x 3 560 x 4 672 x 5 448 x 6 128 x 7
1
3 9 27 81 243 729 2187
ii Use the expansion of 1 x 10 to find the value of 0.99810 to 8 dps
1 x 10 1 10 x 45 x 2 120 x 3 210 x 4 252 x 5 210 x 6 120 x 7 45 x8
10 x 9 x10
0.998 1 100.002 450.002 1200.002
10 2 3
0.98017904
27. iii Find the coefficient of x 2 in 2 3x 4 5 x 4
28. iii Find the coefficient of x 2 in 2 3x 4 5 x 4
2 3x 4 5 x 4
2 3 x 44 443 5x 642 5x 2 445x 3 5x 4
29. iii Find the coefficient of x 2 in 2 3x 4 5 x 4
2 3x 4 5 x 4
2 3 x 44 443 5x 642 5x 2 445x 3 5x 4
30. iii Find the coefficient of x 2 in 2 3x 4 5 x 4
2 3x 4 5 x 4
2 3 x 44 443 5x 642 5x 2 445x 3 5x 4
31. iii Find the coefficient of x 2 in 2 3x 4 5 x 4
2 3x 4 5 x 4
2 3 x 44 443 5x 642 5x 2 445x 3 5x 4
coefficient of x 2 26 4 5 34 4 5
2 2 3
32. iii Find the coefficient of x 2 in 2 3x 4 5 x 4
2 3x 4 5 x 4
2 3 x 44 443 5x 642 5x 2 445x 3 5x 4
coefficient of x 2 26 4 5 34 4 5
2 2 3
4800 3840
960
33. iii Find the coefficient of x 2 in 2 3x 4 5 x 4
2 3x 4 5 x 4
2 3 x 44 443 5x 642 5x 2 445x 3 5x 4
coefficient of x 2 26 4 5 34 4 5
2 2 3
4800 3840
960
Exercise 5A; 2ace etc, 4, 6, 7, 9ad, 12b, 13ac, 14ace, 16a, 22, 23